Modal-Centric Field Inversion via Differentiable Proper Orthogonal Decomposition

TL;DR

This paper introduces a modal-centric inversion method that leverages differentiable POD to efficiently solve high-dimensional inverse problems by focusing on dominant flow structures, significantly reducing computational costs.

Contribution

The paper presents a novel differentiable POD-based framework for inverse problems, enabling efficient gradient computation in modal space and improving scalability for high-dimensional systems.

Findings

Efficient gradient computation via differentiable POD.

Reduced computational cost independent of parameter dimension.

Successful application to inverse problems in viscous Burger's equation.

Abstract

Inverse problems in computational physics often require matching high-dimensional spatio-temporal fields, leading to prohibitive computational costs and ill-conditioned optimizations. We introduce modal-centric field inversion (MCFI), a paradigm that reformulates inverse problems in the reduced space of proper orthogonal decomposition (POD) modes rather than the full physical state space. By targeting dominant flow structures instead of point-wise field values, MCFI provides a compact, physically meaningful objective that naturally regularizes the inversion and dramatically reduces computational burden. Central to this framework is the differentiable POD: an adjoint-based method that efficiently computes sensitivities of POD modes with respect to model parameters, enabling gradient-based optimization in the modal space. We demonstrate MCFI on a one and two-dimensional modified viscous Burger's equation, optimizing spatially varying coefficients to match target dynamics through mode-matching. The adjoint formulation achieves computational cost independent of parameter dimension, in contrast to finite-difference approaches that scale linearly. MCFI establishes a foundation for scalable inverse design and model calibration in unsteady, high-dimensional systems.